USAJMO 2010 Question 3

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Let $P,Q,R,S$ be the feet from $Y$ to the lines $AX,BX,AZ,BZ$, respectively. Prove that the acute angle between $PQ$ and $RS$ is half of $\angle XOZ$, where $O$ is the midpoint of $AB$.

Proposed by Titu Andreescu (credit in the supplied notes).